When seeking to visually represent investment statistics, the financial community often uses a portfolio modeling utility described in a 1952 academic paper by Nobel Laureate Harry Markowitz. Referring to FIG. 1, Markowitz's paper describes how to create a two-dimensional graph of an investment portfolio by plotting the risk associated with an asset against the return the asset generates. Referring to FIG. 2, a curve called the efficient frontier is superimposed upon the graph. This curve represents all possible portfolios of assets that comprise an efficient portfolio for various levels of risk and return. An asset is any resource, quality or feature whether tangible or non-tangible, economic or non-economic. Portfolios are collections of assets. Each unique portfolio is represented as a dot on the graph.
According to Markowitz's model, an efficient portfolio is one where for any given return, no other portfolio has less risk, and for a given level of risk, no other portfolio provides superior returns. This is more specifically defined as a mean-variance efficient portfolio. Other forms of portfolio efficiency occur when one function is maximized and or another function is minimized. Maximization and minimization are processes for discovering a value that serves the objective of the function. For example, given the integers (0, 1, 2, 3, 4, and 5) a minimization function would simply return 0. Another form of portfolio efficiency may be mode—probability of loss efficient. This would indicate that the returns of a portfolio that are observed most frequently (mode) are denominated by a value indicating the probability that the investor would lose on that investment in whole or in part. The maximum value of this ratio provides an efficient portfolio. A portfolio should only be considered efficient if its maximum or minimization functions serve to match the objectives of the investor.
Markowitz's theory generates a simple graph of the tradeoff between risk and return. The backbone of the model is mean-variance optimization (MVO), which minimizes the variance of a portfolio for a given level of returns. Variance is a measure of dispersion equal to the sum of each possible return's distance from the average expected return, weighted by that return's probability of occurring. Variance measures the variation in the value of that portfolio. The formula for portfolio variance is as follows:
            σ      2        ⁢          (              r        p            )        =            ∑              i        =        1            n        ⁢                  ∑                  j          =          1                m            ⁢                        w          1                ⁢                  w          j                ⁢                  σ          ij                                    where wi and wj are the portfolio weights of assets i and j, respectively, and σij is the covariance between them.        
The other part of the optimization is computationally simple. The expected return E(rp) of a portfolio is easily given by the weighted mean of the expected return of individual stocks.
      E    ⁢          (              r        p            )        =            ∑              i        =        1            n        ⁢                  w        i            ⁢              E        ⁢                  (                      r            i                    )                                    n=the number of stocks in the portfolio,        ri=the return of asset i, and        wi=the portfolio weight of asset i,        such that. Σwi=1        
MVO is also central in determining the asset allocation. The optimization procedure searches for the allocation weights that create the portfolio with the best return to risk ratios. Thus, the efficient frontier implicitly calculates correlation by using the portfolio's internal co-variances to reduce aggregate portfolio variance. According to MVO advocates, reducing variance reduces a portfolio's risk. This is the benefit of diversification. Diversification, a portfolio characteristic commonly associated with risk, is placed under the umbrella of the portfolio variance. The outcome of the MVO process is the most variance-efficient portfolio for the aforementioned given level of return.
The efficient frontier is a collection of portfolios representing the most efficient risk to return ratios, all having been created through MVO. The graphic depiction is a two-dimensional curve. Each portfolio, individually representing different—but equally efficient—risk-return combinations is plotted on the curve. Financial professionals like to use this graphic depiction to guide investors' expectations of potential returns and to estimate the risk necessary to achieve those returns. Most professionally created financial plans today show where an investor's portfolio lies relative to the efficient frontier.
Though revolutionary in its time and helpful for important purposes, Markowitz's theory is ineffective for thorough investment analysis. The model assumes variance should be the standard measure for risk, and while some professional money managers hold that this is true, for many investors variance is not the best measurement.
Although the Markowitz model constitutes the majority of portfolio optimization, it is one of many. Diversified portfolios can be constructed without requiring the consideration of standard deviation or variance.
Other forms of portfolio optimization and resource allocation are available. The differences in these optimization techniques may include the mathematical procedure for solving the optimization problem. Markowitz is normally a quadratic optimization; however other techniques may be used including genetic algorithms, stochastic optimization, meta-heuristics, linear programming and non-linear optimization. Despite the numerous variations, only the Markowitz model is commonly associated with graphical representation.
Other techniques also vary in scope and objective. Portfolios may be optimized on nearly any basis. Portfolio optimization functions deviate from just risk and return. One of the more popular alternatives is an asset—liabilities optimization for pension funds. The pension fund may seek to minimize the net present value of future fund contributions and impose liability constraints of/on the expected future fund withdrawals. This allows the fund manger to ensure that the future liabilities of the fund are met, and only the absolute necessary funding is directed from the operations of the business. Similarly, many optimizations seek to determine a maximum utility of wealth. At the personal level, a wealth utility may mean that an investor really values retiring in ten years with 400,000 dollars in the bank. The investor surely prefers $700,000, but the marginal utility of the extra wealth is not worth the extra time and risk necessary to achieve it. The investor has defined the point of maximum wealth utility. The models create an asset allocation model and require many of the same inputs of the securities such as expected returns, measures of risk and the co-variances. This is a broad overview of popular optimization in portfolio management, more specific and intricate optimization is applied to many portfolios for a variety of economic reasons.
An optimization problem is defined by a set of variables and parameters. These variables and parameters combine to form constraints and objectives. The constraints define a feasible search space. The objectives determine the desirability of each point in the space defined by the constraints, compared to all the other points in that space. Optimization techniques use geometric information defined by the objective and the constraints in order to find the most desirable points in the search space. Optimization problems have a complex taxonomy, but in general terms, they can be divided into linear or nonlinear, convex or non-convex, continuous or discrete (combinatorial) and deterministic or stochastic. Depending on the classification, according to all the categories above, different solution techniques can be applied for solving a specific problem.
Correlation is the measure of the statistical relationship between any two events, observations or occurrences. When a positive incremental change in one asset statistic is always related to an equivalent positive incremental change in the other, those statistics have a perfect positive correlation, or a correlation value of 1. On the other hand, when two asset statistics have a perfect negative correlation (a correlation value of −1), a positive incremental change in one is always related to an equivalent negative incremental change in the other. Correlation values between 1 and −1 represent the spectrum of statistical correlation possible. A correlation value of zero indicates that there is no statistical relationship between the compared asset statistics. These correlation values may also be scaled for different ranges.
The notation is included below to calculate the Pearson correlation coefficient:
We use the symbol r to stand for the correlation.
  r  =                    ∑                  i          =          1                n            ⁢                        (                                    x              i                        -                          x              _                                )                ⁢                  (                                    y              i                        -                          y              _                                )                                              (                                    ∑                              i                =                1                            n                        ⁢                                          (                                                      x                    i                                    -                                      x                    _                                                  )                            2                                )                ⁢                  (                                    ∑                              i                =                1                            n                        ⁢                                          (                                                      y                    i                                    -                                      y                    _                                                  )                            2                                )                        yi=An individual observation of the y variable.    xi=An individual observation of the x variable.     x=An average of the x variables.     y=An average of the y variables.    n=The number of observations.
Correlation is the mathematics behind diversification. Industry regulators agree that basing investment decisions on historical returns is perilous. The SEC recognizes this logic as unsound. Figures and statements referencing historical returns often need be accompanied with such disclaimers as, “Past performance is not indicative of future returns.”
Diversification helps investment managers fulfill their fiduciary responsibility to their customers. The law has evolved to subject their rationale to the “prudent investor rule.”Investment managers have fulfilled their legal duties, as fiduciaries, if their investment decisions—win or lose—would be decisions made by a prudent man. In 1976, the U.S. Department of Labor, the government body responsible for Employee Retirement Income Security Act (ERISA), determined that questions of prudence regarding mutual fund investing would be decided on the merits of diversification, not on the basis of individual stock selection. This mode of thinking seems to be in favor with market regulators, yet practitioners have difficulty following the play list.
Despite the merits of diversification, the dreary simplicity of past performance as an asset selection method makes it a dominant method despite the best efforts of government regulators to curtail its use. Correlation is a more effective asset-management tool. Many common practices of investment selection rely on historical returns. Extrapolating historical returns to predict future returns is a poor if not damaging proxy. Inexperienced investors are especially vulnerable to the luster of historical returns. An investor is susceptible to being carried away with a prospective investment merely because it had a superior historical performance. Forcing an investment decision to comply with the portfolio objectives helps prevent chasing yesterday's best performing assets.
Correlations are more predictable than prices. Asset prices are inclined to mean revert. Markets usually over-swing in both directions. This is in the nature of a boom-bust cycle. If one chooses to use historical correlations as a primary model determinant, the model will have more predictive power that any equivalent model that uses historical returns as a primary determinant.
Predicting returns will always be challenging, so diversification will never cease to be a key consideration. If investors are correct in their assessments of future returns, risk is benign. However, when assessments are incorrect the likelihood of loss is greater. It is at these times of inaccurate assessments that a manager's need for risk management is greatest.
An investor who understands that two assets have performed in a given fashion may undertake a position based on the assets' relationship. (See FIGS. 3-11).
Referring to FIG. 3, there is illustrated a correlation matrix. The correlation matrix consists of eight popular futures contracts observed during the year 2000.
Referring to FIG. 4, the S & P 500 and Russell 2000 moved in opposite directions until mid-March, causing a low historical correlation of 0.379. Normally equities of the same country would be expected to have a correlation closer to 0.75. The correlation will vary depending on the time span and observation frequency.
Referring to FIG. 5, this figure shows the time series graph for the Russell 2000 stock index futures. None of the relationships of the Russell with the other futures contracts listed were strong during this period. Strong correlations may be strong positive or strong negative as the correlation approaches 1 or negative 1.
Referring to FIG. 6, this figure shows the time series graph for the U.S. Dollar Index futures contract. The dollar had strong correlations to most other assets indicating that foreign capital flows during the time period were net positive for the United States.
Referring to FIG. 7, this figure shows the time series graph for the Japanese yen futures contract. The yen has a natural negative correlational component to the dollar in that investors wishing to buy dollars may need to sell yen and in order to buy yen investments held in dollars would have to be sold.
Referring to FIG. 8, this figure shows the time series graph for the Eurodollar futures. The Eurodollar and Treasury bond are both interest rate products but are at opposite ends of the yield curve, over time, their prices act similarly providing a definite positive correlation. On a daily basis the relationship is much more moderate. Differences in magnitude, not so much in the direction provide the differences. The correlation of 0.643 confirms this relationship.
Referring to FIG. 9, this figure shows the time series graph for the 30 year Treasury bond futures. The Treasury bond is strongly linked to the dollar, crude and unleaded. This relationship is in part explained in the fact that foreign investors who choose to purchase Treasury bonds must also buy dollars. This helps to explain the positive correlation of 0.721.
Referring to FIG. 10, this figure shows the time series graph for the Crude oil futures. Crude has a correlation with the dollar of 0.897. This strong positive correlation could be used by an investor to better understand the relationships of these important assets.
Referring to FIG. 11, this figure shows the time series graph for unleaded gasoline futures. Unleaded Gasoline is refined from crude oil; their prices are tightly coupled and rarely deviate. This tight relationship is maintained by arbitrageurs who attempt to profit from relative price differences.
To gain a better understanding of correlation, compare the correlation between any of the two assets to the graphs of those assets taken over the same time period. If that investor believes that the relationship of the securities will break from its given pattern, and they believe that they can predict one or more of the respective directions of the pair, the likelihood of a loss is smaller than the likelihood of a loss that would be normally associated with a similar position taken in two assets with an unknown relationship. To incur a loss on the trade the investor must be correct about the decoupling of the assets relationship but be wrong about the magnitudes and/or directions of the individual moves. Alternatively, the investor would substantially break even if the relationship does not decouple. In short, the investor's risk is reduced by correctly forecasting an additional element of analysis; the assets relationship or correlation. Investors who repeat this process for other assets with no relationships or an insignificant relationship to the first pair are able to deliberately build significantly diversified portfolios. Investors may use the correlations of assets' relationships to further isolate and determine the risks that they wish to take or avoid.
The world is filled with different risks. A more desirable model can permit greater breadth of measures, definitions and classifications. MVO has limitations as to how risk can be defined. Nonetheless, MVO has evolved to accommodate semi-variance. Semi-variance is probably a better measure of true risk. Where as variance measures the change associated with the prices of a security, semi-variance measures the variability of price only when the price has a negative movement. This prevents positive price movements from adding to the risk level. While semi-variance moves in the right direction to accurately account for risk, risk contains elements that do not readily lend themselves to one simple quantification. Practitioners may prefer to use multiple measures or a custom measure of risk designed to match their own needs.
Entities, engaged in investments and risk management, are exposed to many different categories of risk. These include:                investment risk: the risk necessary to take or hold a position in the markets.        market risk: the risk given by the broad market or general economic conditions.        credit risk: a measure of the borrower's financial ability to repay debt.        business risk: the risk associated with the management of a particular company in a particular industry and that company's uncertainty of their future earnings.        currency risk: the potential for an investment to be impacted by a change in the exchange rate of any two countries including a direct impact when an investment is denominated in a depreciated foreign currency or indirect when an investment in another asset (i.e. an export business) is damaged when the earnings of that concern fall because of an appreciation of the home currency.        political risk: the potential that changes in legislation or party politics may disrupt the investment expectations or increase the uncertainty of the future.        country risk: risk that changes in the social, economic or political landscape of any one country will affect an investment directly or indirectly.        interest rate risk: measures the sensitivity of an asset to changes in interest rates.        liquidity risk: the risk that buyers or sellers will not be present at the time you wish to trade. Thus, buyers only find sellers at higher prices and sellers only find buyers at lower prices.        counter-party risk: the risk that the counter party to any trade will go bankrupt, refuse to indemnify or not recognize the trade as legitimate.        
Some of these risks can be diversified away by spreading out capital to a broad spectrum of similarly denominated securities. For example, to reduce country risk, one normally seeks to invest capital among many countries with varying geopolitical attributes. Other times these risks can be directly “hedged.” For example, if a portfolio manager expects a particular company's stock to outperform the market, yet is not willing to bear the risk of being in the market, the manager may purchase that stock and sell an equal amount in a market-index contract (such as the S & P 500). In both scenarios, the manager is attempting to achieve a balance.
Visual representation of an asset allocation is largely the domain of the pie chart or a price chart. A pie chart shows the portfolio weights while a price chart shows the performance. A pie chart contains very little information and provides no information as to the internal dynamics of a portfolio. Correlations can be individually graphed in a manner showing a time sequential graphs of one or two assets and a correlation or a regression. (FIGS. 10,11) But a greater visualization platform would foster better understanding and empower individual and professional investors to make better decisions.
In order to better capitalize on diversification, the investor would prefer to be able to visualize and manipulate the individual assets in the portfolio—a benefit they do not receive from the efficient frontier, any asset allocation model or any present form of portfolio optimization. The time has come to broaden our conception of risk, refine our notions of diversification, and reveal portfolio composition in a lucid graphical form.